On D so that x^2 - Dy^2 represents m and -m and not -1

John P. Robertson

National Council on Compensation Insurance, USA

Abstract: For $m = 25$, $100$, $p$, $2p$, $4p$, or $2 p^2$, where $p$ is prime, we show that there is at most one positive nonsquare integer $D$ so that the form $x^2 - D y^2$ primitively represents $m$ and $-m$ and does not represent $-1$. We give support for a conjecture that for any $m > 1$ not listed above, there are infinitely many $D$ so that the form $x^2 - Dy^2$ primitively represents $m$ and $-m$ and does not represent $-1$.